Remark 3.3 (Equilibrium condition as a sequential limit of smooth fields) Consider a stress field such that \(\Sigma  \in \mathcal {B}(\Omega ,\operatorname{Sym})\) , with \(\sigma \)  bounded in \(\Omega \) , and a bending strain field (as considered above) with \(\lambda \)  and \(\gamma \)  bounded in \(\Omega \)  and \(S\) , respectively. We let the moment field to be a general distribution \(M \in \mathcal {D}^{\prime }(\Omega ,\operatorname{Sym})\) . We also assume \(F=0\) . Then, according to Lemma REF , there exist \(\Phi \in \mathcal {B}(\Omega )\) , such that \(\nabla \Phi \)  is continuous across \(S\)  and \(\Sigma =\mathbb {A} \nabla \nabla \Phi \) , and \(W\in \mathcal {B}(\Omega )\)  continuous across \(S\) , such that \(\Lambda =\nabla \nabla W\) . The non-smooth fields \(W,\Phi \) , and \(M\)  can be seen as the weak limits of sequences of smooth fields \(w_n\) , \(\phi _n\) , and \(m_n\)  such that \(\nabla w_n\)  and \(\nabla \nabla \phi _n\)  converge to \(\nabla W\)  and \(\nabla \Phi \)  weakly in \(L^2\)  and \(m_n\)  converges to \(M\)  in the distributional sense [1]}. The stress field \(\Sigma \)  is the \(L^2\)  limit of the sequence of smooth symmetric tensor fields \(\sigma _n=\mathbb {A}\nabla \nabla \phi _n.\)  The equilibrium condition, in terms of the smooth fields \(m_n\) , \(\sigma _n\) , and \(w_n\) , is \(\operatorname{div}\operatorname{div}m_n - [\phi _n,w_n]=0 ~\text{in}~\Omega \) .
Recall that \([\phi _n,w_n]\)  converges to \([\Phi ,W]\)  and \(\operatorname{div}\operatorname{div}m_n\)  converges to \(\operatorname{Div}\operatorname{Div}M\)  in the sense of distributions, see Remark REF . Therefore for weak fields, with the given regularity (bounded densities of stress and bending strain fields and no concentration in stress), equilibrium condition (REF ) can be interpreted as the weak sequential limit of a sequence of equilibrium conditions with smooth fields.

